Group #22626897 — subgroups

Subgroups — C₂×#17989646 #22626897

Order 9344.

368 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 202 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17989646	C₂	1	8	yes	maximal, direct product factor
2	C₂×#14556666	C₂	1	2	yes	maximal, complemented
3	C₂×#14556665	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14556664	C₂	1	2	yes	maximal, complemented
5	C₂²×#11924031	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14556599	C₂	1	1	yes	maximal, characteristic
7	#14556666	C₂²	1	8	yes	complemented
8	#14556665	C₂²	1	4	yes	complemented
9	#14556664	C₂²	1	8	yes	complemented
10	C₂×#11924031	C₂²	1	1	yes	characteristic
11	C₂×#11924031	C₂²	1	1	yes	characteristic
12	C₂×#11924031	C₂²	1	2	yes	—
13	C₂×#11924031	C₂²	1	4	yes	complemented
14	C₂×#11924028	C₄	1	1	yes	characteristic
15	#14556599	C₄	1	4	yes	—
16	#14556599	C₂²	1	4	yes	—
17	C₂×#11924011	C₄	1	2	yes	—
18	C₂²×#10596831	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₇₃	C₂²	1	2	yes	—
20	C₂×M₁₆×C₇₃	C₄	1	1	yes	characteristic
21	#11924031	C₂³	1	2	yes	—
22	#11924031	C₂³	1	2	yes	—
23	#11924031	C₂³	1	4	yes	—
24	#11924028	C₈	1	4	yes	—
25	#11924028	C₂×C₄	1	4	yes	—
26	C₂×Dic₁₄₆	C₈	1	1	yes	characteristic
27	C₂×D₂₉₂	C₈	1	1	yes	characteristic
28	C₂×C₂²⋊D₇₃	C₈	1	2	yes	—
29	C₂×C₄×D₇₃	C₂×C₄	1	2	yes	—
30	#11924011	C₂×C₄	1	8	yes	—
31	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
32	C₂×#10596831	C₂×C₄	1	2	yes	—
33	C₂×#10596831	C₂³	1	1	yes	characteristic
34	C₂×#10596831	C₂³	1	2	yes	—
35	C₈×D₇₃	C₂×C₄	1	4	yes	—
36	C₈×D₇₃	C₂³	1	4	yes	—
37	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
38	M₁₆×C₇₃	C₂×C₄	1	4	yes	—
39	C₂×C₈×C₇₃	C₂×C₄	1	2	yes	—
40	C₂³×C₇₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×D₇₃	C₂×C₈	1	2	yes	—
42	C₂×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×C₇₃	C₂×C₈	1	2	yes	—
44	C₂×C₄×C₇₃	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²⋊D₇₃	C₂×C₈	1	8	yes	—
47	C₂×Dic₇₃	C₂×C₈	1	2	yes	—
48	D₂₉₂	C₂×C₈	1	4	yes	—
49	C₄×D₇₃	C₂×C₈	1	4	yes	—
50	C₄×D₇₃	C₂²×C₄	1	4	yes	—
51	Dic₁₄₆	C₂×C₈	1	4	yes	—
52	C₈×C₇₃	C₂²×C₄	1	4	yes	—
53	#10596831	C₂²×C₄	1	1	yes	characteristic
54	#10596831	C₂²×C₄	1	2	yes	—
55	#10596831	C₂⁴	1	1	yes	characteristic
56	C₂²×C₇₃	—	2	2	no	—
57	C₂²×C₇₃	C₂²×C₈	1	2	yes	—
58	C₂²×C₇₃	C₂²×C₈	1	1	yes	characteristic
59	C₂×D₇₃	—	2	4	no	—
60	C₂×D₇₃	C₂²×C₈	1	4	yes	—
61	C₄×C₇₃	C₂²×C₈	1	2	yes	—
62	C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
63	C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
64	Dic₇₃	C₂²×C₈	1	4	yes	—
65	C₂×C₇₃	—	2	2	no	—
66	C₂×C₇₃	#504	1	2	yes	complemented
67	C₂×C₇₃	C₂³×C₈	1	1	yes	characteristic
68	D₇₃	—	2	4	no	—
69	C₂×#504	—	73	1	no	maximal, Sylow
70	C₇₃	C₂×#504	1	1	yes	characteristic, complemented, Sylow
71	C₂×#131	—	73	1	no	—
72	#504	—	73	8	no	—
73	C₂×M₃₂	—	73	1	no	—
74	C₂×M₃₂	—	73	2	no	—
75	C₂²×C₁₆	—	73	1	no	—
76	C₂²×C₁₆	—	73	2	no	—
77	C₂×#41	—	73	1	no	—
78	#131	—	73	4	no	—
79	#131	—	73	4	no	—
80	C₂×M₁₆	—	73	2	no	—
81	C₂×M₁₆	#64408	1	1	yes	characteristic
82	C₂²×C₈	—	73	1	no	—
83	C₂²×C₈	—	73	2	no	—
84	M₃₂	—	73	4	no	—
85	M₃₂	—	73	8	no	—
86	C₂×C₁₆	—	73	1	no	—
87	C₂×C₁₆	—	73	1	no	—
88	C₂×C₁₆	—	73	2	no	—
89	C₂×C₁₆	—	73	4	no	—
90	C₂×C₁₆	—	73	8	no	—
91	#41	—	73	4	no	—
92	#41	—	73	4	no	—
93	C₂×Q₈	—	73	1	no	—
94	C₂×D₄	—	73	2	no	—
95	C₂×D₄	—	73	1	no	—
96	C₂²×C₄	—	73	2	no	—
97	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
98	M₁₆	—	73	8	no	—
99	M₁₆	C₂×#64408	1	4	yes	—
100	C₂×C₈	—	73	2	no	—
101	C₂×C₈	—	73	2	no	—
102	C₂×C₈	—	73	1	no	—
103	C₂×C₈	—	73	1	no	—
104	C₂×C₈	—	73	4	no	—
105	C₂×C₈	—	73	4	no	—
106	C₂×C₈	C₂×#64408	1	2	yes	—
107	C₁₆	—	73	2	no	—
108	C₁₆	—	73	2	no	—
109	C₁₆	—	73	4	no	—
110	C₂³	—	73	2	no	—
111	C₂³	#11924036	1	1	yes	characteristic
112	Q₈	—	73	4	no	—
113	D₄	—	73	8	no	—
114	D₄	—	73	4	no	—
115	C₂×C₄	—	73	2	no	—
116	C₂×C₄	—	73	4	no	—
117	C₂×C₄	—	73	4	no	—
118	C₂×C₄	#11924036	1	1	yes	characteristic
119	C₂×C₄	#11924036	1	2	yes	—
120	C₂×C₄	C₂²×#64408	1	2	yes	—
121	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
122	C₈	—	73	1	no	—
123	C₈	—	73	1	no	—
124	C₈	—	73	2	no	—
125	C₈	C₂²×#64408	1	4	yes	—
126	C₂²	—	2	2	no	—
127	C₂²	—	73	4	no	—
128	C₂²	—	146	4	no	—
129	C₂²	C₂×#11924036	1	2	yes	—
130	C₂²	C₂×#11924036	1	1	yes	characteristic
131	C₄	—	73	4	no	—
132	C₄	C₂×#11924036	1	1	yes	characteristic
133	C₄	C₂×#11924036	1	2	yes	—
134	C₄	C₂³×#64408	1	1	yes	characteristic
135	C₂	—	2	2	no	—
136	C₂	—	146	4	no	—
137	C₂	#17989646	1	2	yes	direct product factor
138	C₂	C₂²×#11924036	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989646

C₂

yes

maximal, direct product factor

C₂×#14556666

C₂

yes

maximal, complemented

C₂×#14556665

C₂

yes

maximal, characteristic, complemented

C₂×#14556664

C₂

yes

maximal, complemented

C₂²×#11924031

C₂

yes

maximal, characteristic, complemented

C₂×#14556599

C₂

yes

maximal, characteristic

#14556666

C₂²

yes

complemented

#14556665

C₂²

yes

complemented

#14556664

C₂²

yes

complemented

C₂×#11924031

C₂²

yes

characteristic

C₂×#11924031

C₂²

yes

characteristic

C₂×#11924031

C₂²

yes

—

C₂×#11924031

C₂²

yes

complemented

C₂×#11924028

C₄

yes

characteristic

#14556599

C₄

yes

—

#14556599

C₂²

yes

—

C₂×#11924011

C₄

yes

—

C₂²×#10596831

C₂²

yes

characteristic

C₂×C₈×D₇₃

C₂²

yes

—

C₂×M₁₆×C₇₃

C₄

yes

characteristic

#11924031

C₂³

yes

—

#11924031

C₂³

yes

—

#11924031

C₂³

yes

—

#11924028

C₈

yes

—

#11924028

C₂×C₄

yes

—

C₂×Dic₁₄₆

C₈

yes

characteristic

C₂×D₂₉₂

C₈

yes

characteristic

C₂×C₂²⋊D₇₃

C₈

yes

—

C₂×C₄×D₇₃

C₂×C₄

yes

—

#11924011

C₂×C₄

yes

—

C₂×#10596831

C₂×C₄

yes

characteristic

C₂×#10596831

C₂×C₄

yes

—

C₂×#10596831

C₂³

yes

characteristic

C₂×#10596831

C₂³

yes

—

C₈×D₇₃

C₂×C₄

yes

—

C₈×D₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

M₁₆×C₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

C₂×C₄

yes

—

C₂³×C₇₃

C₂×C₈

yes

characteristic

C₂²×D₇₃

C₂×C₈

yes

—

C₂×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₈

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂²⋊D₇₃

C₂×C₈

yes

—

C₂×Dic₇₃

C₂×C₈

yes

—

D₂₉₂

C₂×C₈

yes

—

C₄×D₇₃

C₂×C₈

yes

—

C₄×D₇₃

C₂²×C₄

yes

—

Dic₁₄₆

C₂×C₈

yes

—

C₈×C₇₃

C₂²×C₄

yes

—

#10596831

C₂²×C₄

yes

characteristic

#10596831

C₂²×C₄

yes

—

#10596831

C₂⁴

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

C₂²×C₈

yes

—

C₂²×C₇₃

C₂²×C₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

C₂²×C₈

yes

—

C₄×C₇₃

C₂²×C₈

yes

—

C₄×C₇₃

C₂²×C₈

yes

characteristic

C₄×C₇₃

C₂³×C₄

yes

characteristic

Dic₇₃

C₂²×C₈

yes

—

C₂×C₇₃

—

C₂×C₇₃

#504

yes

complemented

C₂×C₇₃

C₂³×C₈

yes

characteristic

D₇₃

—

C₂×#504

—

maximal, Sylow

C₇₃

C₂×#504

yes

characteristic, complemented, Sylow

C₂×#131

—

#504

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

#64408

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#64408

yes

characteristic

M₁₆

—

M₁₆

C₂×#64408

yes

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂×#64408

yes

—

107

C₁₆

—

108

C₁₆

—

109

C₁₆

—

110

C₂³

—

111

C₂³

#11924036

yes

characteristic

112

Q₈

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

#11924036

yes

characteristic

119

C₂×C₄

#11924036

yes

—

120

C₂×C₄

C₂²×#64408

yes

—

121

C₂×C₄

C₂²×#64408

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

C₂²×#64408

yes

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

146

—

129

C₂²

C₂×#11924036

yes

—

130

C₂²

C₂×#11924036

yes

characteristic

131

C₄

—

132

C₄

C₂×#11924036

yes

characteristic

133

C₄

C₂×#11924036

yes

—

134

C₄

C₂³×#64408

yes

characteristic

135

C₂

—

136

C₂

—

146

—

137

C₂

#17989646

yes

direct product factor

138

C₂

C₂²×#11924036

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#64408
commutator	C₂×C₇₃	C₂³×C₈
Frattini	C₄	C₂³×#64408
Fitting	C₂×M₁₆×C₇₃	C₄
socle	C₂²×C₇₃	C₂²×C₈
hypercenter	C₂×M₁₆	#64408
hypocenter	C₇₃	C₂×#504
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₂×C₄

C₂²×#64408

commutator

C₂×C₇₃

C₂³×C₈

Frattini

C₄

C₂³×#64408

Fitting

C₂×M₁₆×C₇₃

C₄

socle

C₂²×C₇₃

C₂²×C₈

C₂×M₁₆

C₇₃

G ▷ C₂×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
1. #17989646	C₂	direct (N × H)
2. C₂×#14556666	C₂	semidirect
3. C₂×#14556665	C₂	semidirect
4. C₂×#14556664	C₂	semidirect
5. C₂²×#11924031	C₂	semidirect
7. #14556666	C₂²	semidirect
8. #14556665	C₂²	semidirect
9. #14556664	C₂²	semidirect
13. C₂×#11924031	C₂²	semidirect
66. C₂×C₇₃	#504	semidirect
70. C₇₃	C₂×#504	semidirect
137. C₂	#17989646	direct (N × H)

H ≅ G/N

type

1. #17989646

C₂

direct (N × H)

2. C₂×#14556666

C₂

semidirect

3. C₂×#14556665

C₂

semidirect

4. C₂×#14556664

C₂

semidirect

5. C₂²×#11924031

C₂

semidirect

7. #14556666

C₂²

semidirect

8. #14556665

C₂²

semidirect

9. #14556664

C₂²

semidirect

13. C₂×#11924031

C₂²

semidirect

66. C₂×C₇₃

#504

semidirect

70. C₇₃

C₂×#504

semidirect

137. C₂

#17989646

direct (N × H)

Subgroups — C₂×#17989646 #22626897

Conjugacy classes of subgroups

Complements

Subgroups — C2×#17989646 #22626897

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#17989646 #22626897